Problem Set - Continuous functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Continuous functions

	Let $f:ℝ\to ℝ$ be such that $f$ is continuous at $x=0$ and if $x,y\in ℝ$ then $f(x+y)=f\left(x\right)f\left(y\right)$. Show that if $a\in ℝ$ then $f$ is continuous at $x=a$.
	Let $f:{ℝ}_{>0}\to ℝ$ be such that $f$ is continuous at $x=1$ and if $x,y\in {ℝ}_{>0}$ then $f\left(xy\right)=f\left(x\right)+f\left(y\right)$. Show that if $a\in {ℝ}_{>0}$ then $f$ is continuous at $x=a$.
	Let $I$ be an interval in $ℝ$. Let $f:I\to ℝ$ be continous. Show that the function $\left|f\right|:I\to ℝ$ given by $\left|f\right|\left(x\right)=\left|f\right(x\left)\right|$ is continuous.
	Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ and $g:I\to ℝ$ be continous. Show that the function $\max (f,g):I\to ℝ$ given by $\max (f,g)\left(x\right)=\max \left(f\right(x),g(x\left)\right)$ is continuous.
	(Thomae's function) Let $f:[0,1]\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}\frac{1}{n},& \text{if}\frac{m}{n}\in \mathbb{Q}\text{is reduced,}\hfill \\ 0,& \text{if}x\notin \mathbb{Q}.\hfill \end{array}$ Show that (a)   If $a\notin \mathbb{Q}$ then $f$ is continuous at $x=a$, and (b)   If $a\in \mathbb{Q}$ then $f$ is not continuous at $x=a$.
	Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ and $g:I\to ℝ$ be continous. Show that the function $\min (f,g):I\to ℝ$ given by $\min (f,g)\left(x\right)=\min \left(f\right(x),g(x\left)\right)$ is continuous.
	Let $a\in ℝ$ and let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}ax,& \text{if}x\le 0,\hfill \\ \sqrt{x},& \text{if}x>0.\hfill \end{array}$ Show that $f$ is continuous.
	Is the function $f:ℝ\to ℝ$ given by $f\left(x\right)=x$ uniformly continuous?
	Is the function $f:(0,1)\to ℝ$ given by $f\left(x\right)=\frac{1}{x}$ uniformly continuous?
	Is the function $f:({10}^{-4},1)\to ℝ$ given by $f\left(x\right)=\frac{1}{x}$ uniformly continuous?
	Is the function $f:(0,1)\to ℝ$ given by $f\left(x\right)=x^2$ uniformly continuous?
	Is the function $f:[-1,1]\to ℝ$ given by $f\left(x\right)=\sqrt{1-x^2}$ uniformly continuous?
	Is the function $f:(1,\infty )\to ℝ$ given by $f\left(x\right)=\log x$ uniformly continuous?
	Is the function $f:(0,\infty )\to ℝ$ given by $f\left(x\right)=\log x$ uniformly continuous?
	Let $f:ℝ\to ℝ$ be the function given by $f\left(x\right)=\frac{x}{(1+|x\left|\right)}$. Show that (a)   $f$ is continuous, (b)   $f$ is uniformly continuous, (c)   $\sup \left(f\right(ℝ\left)\right)=1$, (d)   There does not exist $x\in ℝ$ such that $f\left(x\right))=1$, (e)   $\inf \left(f\right(ℝ\left)\right)=-1$, (d)   There does not exist $y\in ℝ$ such that $f\left(y\right))=-1$.
	Show that the function $f:ℝ\to ℝ$ given by $f\left(x\right)=x^3-6x+3$ has exactly 3 roots.
	Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ be a continuous function. Prove that $f\left(I\right)$ is an interval.
	Let $I$ and $J$ be intervals in $ℝ$ and let $f:I\to J$ be a surjective strictly monotonic continuous function. Prove that the inverse function $g:J\to I$ exists and is strictly monotonic and continuous.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)